\documentclass[11pt]{amsart}
\usepackage{geometry}
\geometry{letterpaper}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{siunitx}
\usepackage{tikz}
\usepackage{lscape}
\usetikzlibrary{dsp,fit,positioning}

\input{../util/airfoils.tex}
\input{../util/wing.tex}
\input{../util/coordinate_systems.tex}
\input{../util/control.tex}

\newcommand{\qhat}{\hat{q}}
\newcommand{\cbar}{\bar{c}}
\newcommand{\qbar}{\bar{q}}
\newcommand{\cmd}{\mathrm{cmd}}
\newcommand{\ff}{\mathrm{ff}}
\newcommand{\eff}{\mathrm{eff}}
\newcommand{\app}{\mathrm{app}}
\newcommand{\wind}{\mathrm{wind}}
\newcommand{\kite}{\mathrm{kite}}
\newcommand{\nom}{\mathrm{nom}}
\newcommand{\aero}{\mathrm{aero}}
\newcommand{\geom}{\mathrm{geom}}

\begin{document}

\section{Servo model}
Expressing everything in terms of shaft angle $\theta$, which is $\theta_m / g$.
\begin{eqnarray}
\dot{\theta}_{\mathrm{ref}} &=& \omega_{\mathrm{ref}}
(\theta_{\mathrm{cmd}} - \theta_{\mathrm{ref}}) \\
v &=& k_p (\theta_{\mathrm{ref}} - \theta) +
k_d (\dot{\theta}_{\mathrm{ref}} - \dot{\theta}) \\
i &=& (v - k_{\tau} g \dot{\theta}) / Z \\
I_{\mathrm{servo}}\, g \ddot{\theta} &=& k_{\tau} i - b \dot{\theta} / g +
\tau_{\mathrm{ext}} / g \\
\tau_{\mathrm{ext}} &=& \frac{1}{2} \rho v^2 A c C_{m_{\delta_i}} \theta
\end{eqnarray}
$\omega_{\mathrm{ref}} = 100$ rad/s, $k_p = 1000$ V/rad, $k_d = 100$
V/rad-s, $k_{\tau} = 0.239$ N-m/A, $g = 160$,
$I_{\mathrm{servo}}= 2.3 \times 10^{-4}$ kg-m$^2$.
$Z = \sqrt{R^2 + (g \dot{\theta} L)^2}$.
$R = 0.886$.  The friction constant $b$ is between 70 and 170
N-m/(rad/s) of shaft velocity.  For the ailerons, $C_{m_{\delta_a}} =
0.0026$.

\begin{equation}
\frac{\Theta_{\mathrm{ref}}(s)}{\Theta_{\mathrm{cmd}}(s)} =
\frac{\omega_{\mathrm{ref}}}{s + \omega_{\mathrm{ref}}}
\end{equation}

\begin{equation}
V(s) = (k_p + k_d s) (\Theta_{\mathrm{ref}}(s) - \Theta(s))
\end{equation}

\begin{equation}
I_{\mathrm{servo}}\, g s^2 \Theta(s) =
k_{\tau} (k_p + k_d s) (\Theta_{\mathrm{ref}}(s) - \Theta(s)) / Z -
k_{\tau}^2 g s \Theta(s) / Z
\end{equation}

\begin{equation}
\frac{\Theta(s)}{\Theta_{\mathrm{cmd}}(s)} =
\frac{\omega_{\mathrm{ref}}}{s + \omega_{\mathrm{ref}}}
\cdot
\frac{k_{\tau} (k_p + k_d s)}
{g Z I_{\mathrm{servo}} s^2 + (k_{\tau} k_d + k_{\tau}^2 g) s +
(k_{\tau} k_p - \frac{Z \bar{q} A c}{g} C_{m_{\delta_i}})}
\end{equation}

\end{document}
